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PROFESSOR: Welcome
back to recitation.

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In this video I'd
like us to practice

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some of the
approximation techniques

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we've learned in the lectures.

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So in this one
specifically, what

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I'd like us to do is estimate
the integral from 0 to pi

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of sin x dx using these
two approximation methods.

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a is going to be using
the trapezoid rule,

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and b is going to be
using Simpson's rule.

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In both cases, I'd like you
to do this for n equal 4,

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and to get you started I have
drawn a little sketch of what

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y equals sin x looks like on
the interval from 0 to pi.

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So I'll give you a while
to work on this and then

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when you're finished
you can come back

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and I'll show you how I do it.

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OK.

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Welcome back.

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Hopefully you were able
to get an approximation

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for both of these.

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Now I'll show you how
I do it and make sure

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that you're doing it
in the correct way.

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So I've done a little bit
of work ahead of time,

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because I didn't want to
have to write everything down

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in the video.

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So if you come over here
to the right a little bit,

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we'll see that
I've actually made

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myself a table of the important
x-values and y-values.

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And if you recall,
what I said is

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we're going to approximate
the integral from 0

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to pi using four sub-intervals.

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So we're interested in
i going from 0 to 4,

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and then we need to subdivide
that interval from 0

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to pi into four
equal sub-intervals.

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So that will be
length pi over 4.

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So x_0 is just 0,
x_1 is pi over 4.

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x_2 is pi over 2,
you get the idea.

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x sub 4 is pi.

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And then what I've done, is to
make it really easy on myself,

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is I've determined what
the y-values are associated

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to that x-value, given
that my function is sin x.

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So I filled in a table of
values for myself right away,

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and then when I want
to use the two rules,

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I can simply come back
and look at this table.

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So the first thing I'm going
to do is the trapezoid rule.

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So I'll come back over
here a little bit.

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So using the trapezoid
rule, let's remember

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what the trapezoid rule is.

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The trapezoid rule is delta
x times, in this case,

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we're going to have y_0 over
2 plus y_1 plus y_2 plus y_3

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plus y_4 over 2.

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So let me actually move that.

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So remember, what you get
is you get essentially

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you're averaging the right and
the left hand end point ones.

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So you end up with the furthest
left has a 1/2 coefficient,

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the furthest rights has a 1/2,
all the other coefficients

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are equal to 1.

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So what do we get when we
actually evaluate this?

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When we plug in what we have
for delta x and for all these

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values.

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Delta x, we know
again, it should

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be the length of the
interval divided by n.

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We're sub-dividing equally here.

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So we get pi over 4.

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And then let's
look at our values.

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Well, y_0 and y_4
are both 0, so I'm

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going to not even put those in.

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You'll see 0 there, 0 there,
so these two are both 0.

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So I just have to
substitute in these values.

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y_1, y_2, and y_3.

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So y sub 1 is root 2
over 2, y sub 2 is 1,

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and y sub 3 is root 2 over 2.

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So I should get root 2 over
2 plus 1 plus root 2 over 2.

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And if you want to simplify a
little bit, you can do that.

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Root 2 over 2 plus root
2 over 2, is root 2.

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Hopefully you got something
that looked like this.

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And I gotta tell you at
this point, I'm stopping.

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Because I don't want to bother
to simplify any more than this.

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But the main point is, I want
to make sure we understand,

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once we have this method how
to substitute everything in.

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So this first one was just
using the trapezoid rule

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to approximate the integral.

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Now, what does that actually
mean in terms of the graph?

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Let's go back to the
graph and let's look

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at what that actually means.

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I'm going to get
another color for this.

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So in the case of the trapezoid
rule, what that means,

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I'm sub-dividing the
interval like this

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and the trapezoid
rule, remember,

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connects the two y-values.

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Consecutive y-values here.

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And it's giving you the
area of each of those.

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So we found the blue, we
found the blue shaded area

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with the trapezoid rule.

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So just to recall, that's
actually what we did.

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This is a fairly
good approximation,

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looks like in this case, of
what the actual integral is.

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So now let's do Simpson's rule.

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And I'll come over to
the right of the table

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to do Simpson's rule.

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Now notice I can
do Simpson's rule

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because n is an even number.

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I have to have n even in
order to do Simpson's rule.

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So just to remind us
what Simpson's rule is,

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Simpson's rule is
delta x over 3 and then

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I have these funny coefficients,
which at some point

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we will explain.

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I have a 1 in front of
y_0, a 4 in front of y_1,

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2 in front of y_2, a 4
in front of y_3, and a 1

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in front of the y sub 4.

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So that's exactly
the coefficients

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for Simpson's rule.

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And you saw in class
why this is a 2.

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You'll see in another
recitation video

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why we end up
getting the 1, 4, 1.

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And those two things add
up, 1, 4, 1; 1, 4, 1.

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And where the 3 comes from.

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We'll show all of
that in another video.

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So let's fill in what we have.

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Well delta x is pi over 4 so
I get pi over 4 times 1/3,

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so I get pi over 12.

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y_0 is again 0.

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y_1, come back here,
it was root 2 over 2.

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Root 2 over 2 times
4 is 2 root 2.

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y_2, if you remember, was 1.

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So I get 2 times 1 is 2.

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And then I have the
same two values.

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Because this is a-- this is
symmetric about the y_2 value.

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So I get another 2
root 2, and another 0.

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So if I simplify all this, I get
pi over 12-- oops-- pi over 12

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and then I get 4 root 2 plus 2.

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And so that we can maybe see
a little bit more what it,

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how it compares, we can
simplify this a little bit.

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I'm going to put the 1 in
front, 1 plus 2 root 2.

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Now, if you wanted to actually
do a comparison of those two

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values, and then compare
that to the actual integral,

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you'd want to evaluate
the actual integral

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and then maybe look at what
these two are on a calculator.

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But, I just wanted
to make sure we

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knew how to plug in the y-values
and the appropriate delta

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x to the appropriate formula.

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So, again, what we
were trying to do

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is estimate the integral for
one that we actually know,

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so we could do some
comparison if we wanted.

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And see how these
numerical methods,

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or these approximations
actually work.

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So we did trapezoid rule,
we had the formula up there.

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We did Simpson's rule and
the formula's right here.

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And when we were doing this,
the thing that made it simpler

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is at the very beginning I
made a nice table of values.

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So when you're solving
these types of problems

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that might be something
you want to think

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about doing right at
the very beginning

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to make things a little
simpler for yourself.

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And I think I'll stop there.